Download Network structural properties mediate the stability of mutualistic

Ecology Letters, (2008) 11: 208–216
doi: 10.1111/j.1461-0248.2007.01137.x
LETTER
Network structural properties mediate the stability
of mutualistic communities
Toshinori Okuyama and
J. Nathaniel Holland*
Department of Ecology and
Evolutionary Biology, Rice
University, Houston, TX 77005,
USA
*Correspondence: E-mail:
[email protected]
Abstract
Key advances are being made on the structures of predator–prey food webs and
competitive communities that enhance their stability, but little attention has been given
to such complexity–stability relationships for mutualistic communities. We show, by way
of theoretical analyses with empirically informed parameters, that structural properties
can alter the stability of mutualistic communities characterized by nonlinear functional
responses among the interacting species. Specifically, community resilience is enhanced
by increasing community size (species diversity) and the number of species interactions
(connectivity), and through strong, symmetric interaction strengths of highly nested
networks. As a result, mutualistic communities show largely positive complexity–stability
relationships, in opposition to the standard paradox. Thus, contrary to the commonlyheld belief that mutualismÕs positive feedback destabilizes food webs, our results suggest
that interplay between the structure and function of ecological networks in general, and
consideration of mutualistic interactions in particular, may be key to understanding
complexity–stability relationships of biological communities as a whole.
Keywords
Asymmetry, community structure, complexity, connectivity, degree, food web, interaction strength, mutualistic network, nestedness, species diversity, stability.
Ecology Letters (2008) 11: 208–216
INTRODUCTION
One of the most enduring problems in ecology is
understanding how the stability of a community is influenced by the structural properties that arise from interactions among species comprising the community. Through
the later portion of the last century, the commonly-held
belief was that complexity, as often inferred by species
diversity, begets stability (McCann 2000; Ives & Carpenter
2007). MayÕs (1974) results called into question such positive
complexity–stability relationships, as increasing species
diversity and the number and strength of species interactions of randomly structured communities tended to
diminish stability. Since then, much consideration has been
given to complexity–stability relationships of predator–prey
food webs and competitive communities (McCann 2000;
Ives & Carpenter 2007). Insights are being made into the
community structures and features of species interactions
that confer stability in food webs and competitive systems,
including, for example, community size, compartments,
structural asymmetries, connectivity, interaction strengths,
foraging biology and body size (McCann et al. 1998; Berlow
2007 Blackwell Publishing Ltd/CNRS
1999; Ives et al. 1999; Lehman & Tilman 2000; Ives &
Hughes 2002; Neutel et al. 2002; Kondoh 2003; Krause et al.
2003; Woodward et al. 2005; Rooney et al. 2006).
On the other hand, little attention has been given to
complexity–stability relationships of mutualistic communities involving beneficial (rather than antagonistic) interactions among species. Lotka-Volterra models with linear
functional responses suggest that mutualistic interactions are
either unstable, leading to unbounded population growth
due to never-ending positive feedback, or stable only when
their interaction strengths are weak and ⁄ or asymmetric
(Gause & Witt 1935; Bascompte et al. 2006a). In this regard,
MayÕs (1974) linear models suggest that mutualism destabilizes predator–prey food webs. We now recognize, however,
that when nonlinear functional responses are incorporated
into models, mutualistic systems are not inherently unstable
(May 1981; Holland et al. 2002, 2006; Thompson et al.
2006). Although studies have begun to examine coexistence
and diversity–stability relationships of mutualistic communities (Dodds & Henebry 1996; Pachepsky et al. 2002;
Hoeksema & Kummel 2003; Zhang 2003), few have
explicitly examined how particular structures affect the
Letter
stability of mutualistic communities involving appropriate
nonlinear functional responses.
We have been limited in our ability to investigate how
structural properties influence the dynamics and stability of
mutualistic communities, largely due to the lack of a general
means with which to characterize their community structures, like that of a food web for predator–prey interactions.
Recently, however, the extension of network theory to
mutualistic communities [and food webs (Dunne et al. 2002;
Williams et al. 2002; Garlaschelli et al. 2003; Pascual &
Dunne 2006)] has provided such a means with which to
describe their structural properties arising from interactions
(edges) among species (nodes) comprising a community
(network). Bipartite networks have been employed to depict
mutualistic communities composed of plant–pollinator,
plant–seed disperser, plant–ant, cleaner–client fishes and
anemonefish–sea anemone interactions (Bascompte et al.
2003, 2006a; Jordano et al. 2003; Vázquez 2005; Guimarães
et al. 2006, 2007; Lewinsohn et al. 2006; Ollerton et al. 2007;
Vázquez et al. 2007). In ecological terms, depicting such
mutualistic communities as bipartite networks is simply a
way of describing their two trophic-level food webs, as
nearly all mutualisms are mediated by their consumer–
resource interactions (Holland et al. 2005).
Through the aid of network theory, mutualistic communities are now recognized to entail structural properties of
community size, nestedness, degree and the strength and
(a)symmetry of species interactions (Bascompte et al. 2003,
2006a; Jordano et al. 2003; Vázquez 2005; Guimarães et al.
2006, 2007; Lewinsohn et al. 2006; Ollerton et al. 2007;
Vázquez et al. 2007). Community size is the number of
species comprising a community (species diversity). Nestedness is the topology of an ordered bipartite network
describing non-random interactions among species. Mutualistic networks are often nested, entailing a core of species
with many interactions among themselves, species with few
interactions interacting with proper subsets of species with
many interactions and few if any interactions among species
with few interactions (Bascompte et al. 2003; Guimarães
et al. 2006, 2007; Lewinsohn et al. 2006; Ollerton et al. 2007).
Degree is the number of links a species has with other
species. For mutualistic networks (and food webs), degree
often follows a power distribution, P(k) kc, where k is
degree and c is the degree exponent characterized by
negative values (c < )1; Dunne et al. 2002; Jordano et al.
2003; Vázquez 2005). As c increases (i.e. less negative
values), the number of links per species and community
connectivity increase. Finally, the symmetry of interaction
strengths describes the congruence between the pairwise per
capita benefits of interacting species. Using frequency (or
relative frequency) of interaction as an estimate of interaction strength, mutualistic communities have been suggested
to be characterized by many weak, few strong and
Stability of mutualistic networks 209
asymmetric interaction strengths (Memmott 1999; Bascompte et al. 2006a; Vázquez et al. 2007).
Although progress is being made in elucidating the
structural properties of ecological networks, we are just
beginning to grasp their functional consequences for the
stability of dynamically nonlinear systems (Proulx et al.
2005). With the recent identification of some general
structural properties, studies can begin to more thoroughly
examine complexity–stability relationships of mutualistic
communities involving appropriate nonlinear functional
responses, which in turn may provide a foundation upon
which to develop general theory and guide empiricism.
Here, we take a step towards understanding complexity–
stability relationships of mutualistic communities by examining how their stability varies with the structural properties
of community size, nestedness, degree and the strength and
symmetry of species interactions.
DYNAMIC MODEL OF MUTUALISTIC
COMMUNITIES
To examine relationships between the structural properties
and stability of mutualistic communities, we developed a
dynamic model that moves beyond Lotka-Volterra models
with linear functional responses to one that incorporates the
biological feature of mutualismÕs benefits on a species
saturating with the population densities of others (May
1981; Holland et al. 2002, 2006; Thompson et al. 2006). As
they are often bipartite in nature, we describe mutualistic
communities as being composed of two separate groups of
species (partite M, partite A, e.g. plants, pollinators), for
which NM and NA are the numbers of mutualistic species in
partites M and A, respectively, and Mi and Aj represent the
densities of species i and j of each partite:
NA
X
cij Aj Mi
dMi
P
¼ ri M i Si Mi2 þ
1 þ
dt
a
I½aik >0 Ak
j¼1 ij
ð1Þ
NM
X
dAj
wji Mi Aj
¼ qj Aj Tj A2j þ
P
1
dt
I½bjk >0 Mk
i¼1 bji þ
ð2Þ
k
k
The first of the three terms on the right-hand side of the
equations represent population growth, for which ri and qj
are intrinsic growth rates. The second term modifies
population growth by density-dependent self-limitation,
strengths of which are described by Si and Tj. The third
term represents the beneficial effects of mutualism on
population growth. We simply employed a hyperbolic
functional response to represent that the benefits to
mutualists saturate with the densities of mutualistic species
with which they interact. Interaction strengths are described
by the half-saturation constants (aij and bji) of the functional
2007 Blackwell Publishing Ltd/CNRS
210 T. Okuyama and J. N. Holland
responses, and symmetry of interaction strengths is
described by the similarity in the magnitudes of such
pairwise interaction strengths. Parameters c and w are the
maximum benefits of mutualistic interactions. The summation factor in the denominator of the third term incorporates the total density of species that interact with a
mutualist. The indicator function, I, is used
Pto express the
presence of a species interaction, such that k I½aik >0 Ak for
example, is the total density of mutualistic species that
interact with the ith species of partite M. A summary of the
modelÕs parameters and variables is shown in Table S1.
MODEL ANALYSES
We conducted numerical analyses of the model to examine
how community stability varies with the structural properties
of community size, nestedness, degree and the strength and
symmetry of mutualistic interactions. We varied each
structural property systematically, with the range of their
parameter values informed by empirical estimates (Table S2)
from real mutualistic communities (Bascompte et al. 2006a).
The empirical data show realistic ranges for parameter values
of each structural property, but the parameter values may be
confounded by one another, as any biological mechanism
responsible for one structure (e.g. nestedness, degree distribution) may contribute to and vary with another structure
(e.g. community size). Simulations allowed us to evaluate the
effect of each structural property (holding others constant)
and to evaluate interactions among them, thereby minimizing
any confounding effects of the structural properties. The
range of parameter values for each structure, and default
values for other model parameters, are shown in Table S1.
The model was simulated with initial values randomly
generated from a uniform distribution [domain (1, 2) and
(0.5, 1.5) for Mi and Aj, respectively]. We performed > 100
simulations of the model for each set of analyses of the
structural properties. A number of different metrics may be
employed to measure community stability (McCann 2000;
Ives & Carpenter 2007). Because our model exhibited a
stable equilibrium regardless of parameter values, we
examined stability using community resilience, that is
increases in return rates to the equilibrium following a
perturbation (May 1974; McCann 2000; Ives & Carpenter
2007), as evaluated by the absolute value of the dominant
eigenvalue of the Jacobian matrix at equilibrium (Appendix
S1). In this way, our results are consistent with the
framework of May (1974), and are directly comparable with
his and more recent results on food webs.
Community size
Community size is the total number of species making up a
network, i.e. the sum of mutualistic species NM and NA in
2007 Blackwell Publishing Ltd/CNRS
Letter
the community. Community size of the empirical database
exhibited great variation, ranging from 13 to 951 species
(Table S2). Because most of the empirical networks
clustered towards the lower end of this range, we limited
our analyses of community size to 100 species (NM = 30,
NA = 70), unless otherwise stated.
Degree distribution
Degree, k, is the number of links (interactions) species have
with other species, which for a community of interacting
species often follows a power distribution, P(k) kc, where
c is the degree exponent and is typically a negative value. We
assumed that degree distribution follows a modified power
c
distribution
PNM P(k)
¼ Ck , where k ¼ 1… NM, c < 0, and
c 1
C ¼
is a normalization factor. Thus, the
i¼1 k
domain of the degree distribution is constrained by NM.
Increasing c (i.e. less negative values) leads to an increase in
the average number of species interactions per species and a
decrease in variation among species in their degree. The c
parameter of the empirical database was estimated using a
maximum likelihood approach. To simulate degree, we
assigned the number of species interactions for each species
in partite A by generating random deviates from the
modified power distribution, which were then assigned to
each species in partite A (see Nestedness below).
Nestedness
To simulate nestedness, we ranked species based on their
degree, after degree was first simulated for each species of
partite A. The actual species associations were determined
by a modified multinomial distribution of size NA, Multi(U,
p). Each element of a random vector was constrained to
either 0 for no interaction or 1 for an interaction. For the jth
species of partite A, we set U ¼ Zj, where Zj (the simulated
degree distribution) is the number of species of partite M
with which the jth species of partite A interacts. The
probability vector p is:
p¼
½ðn1 þ 1Þv ; ðn2 þ 1Þv ; ðn3 þ 1Þv ; . . . ; ðnNA þ 1Þv P
;
ðnj þ 1Þv
ð4Þ
j
where ni is the number of species of partite A with which
the ith species of partite M has already formed an
association. Because this procedure was performed sequentially from the most to least connected species, we were able
to manipulate nestedness by varying v. When v = 0, each
mutualistic species forms interactions independent of other
species. Thus, in this algorithm v and nestedness are
positively correlated. For consistency with prior studies
(Bascompte et al. 2003), we report nestedness as
(100 ) T ) ⁄ 100, where T is matrix temperature (Atmar
Letter
Stability of mutualistic networks 211
and Patterson 1993), such that large values represent highly
nested networks. We also calculated nestedness for the
empirical networks to guide parameter values of the
simulations (Table S2).
Strength and symmetry of mutualistic interactions
Mean, variance and correlation of pairwise interaction
strengths were simulated through aij and bji, for which
larger values mean stronger interaction strengths. Symmetric
interaction strengths are described by positive correlations
between aij and bji, whereas asymmetric interaction
strengths are described by negative correlations. Pairwise
interaction strengths were generated from a bivariate
lognormal distribution, with a specified symmetry between
a and b, and mean
of
values and variances
r2
r2LN
2
l ¼ lnðlLN Þ 12 ln 1 þ lLN
¼
ln
1
þ
and
r
.
The
2
l2LN
LN
normal variates were exponentiated to convert them to
lognormal random variables. This creates lognormal samples
with an expected mean and variance of lLN [e.g. E(a)] and
r2LN [e.g. var(a)]. In examining the strengths of interactions,
for simplicity we assumed that the mean and variance of a
and b were the same [E(a) = E(b), var(a) = var(b)]. In
examining the symmetry of interaction strengths, the
correlation between aij and bji were varied through the
correlation parameter, q. For consistency with Bascompte
et al. (2006a), we express asymmetry of interaction strengths
using the equation Aa,b = |a ) b| ⁄ max(a,b), ranging from
0 for perfectly symmetric (i.e. a = b) to 1.0 for highly
asymmetric interactions (i.e. great disparity between values
of a and b).
Figure 1 Resilience of mutualistic communities and their structural
property of community size. Community size is the sum of the
total number of species comprising the mutualistic community.
Resilience, as depicted by the colour code scale, increases with the
absolute value of eigenvalues and colour shading from white to red.
Results are reported for simulation averages.
RESULTS
Regardless of the particular parameter values of the model,
mutualistic communities always exhibited a stable equilibrium and each of the five structural properties contributed
to their resilience, as measured by their transient dynamics in
return rates to the equilibrium after a perturbation.
Resilience increased with community size, whether through
increases in the number of mutualistic species of partite M,
A or both (Fig. 1), thereby indicating that species diversity
does indeed beget the stability of mutualistic communities.
This positive diversity–stability relationship did not depend
on or vary with other community structures. Resilience also
showed a positive relationship with interaction strengths
(a and b), such that strong rather than weak interactions
facilitated stability (Fig. 2). Like species diversity, the
positive relationship between interaction strengths and
resilience did not depend on other community structures.
Although this positive relationship between interaction
strength and resilience remained for both low and high variance in interaction strengths, high variance [var(a) = 1.0,
Figure 2 Community resilience and the strength of mutualistic
interactions. The relationship between resilience and mean
interaction strength is reported for both low and high variance
in interaction strengths. Results are reported for simulation
averages. Note that, for simplicity, we set E(a) = E(b) and
var(a) = var(b) (see Model analyses).
Fig. 2] in interaction strengths had a small, negative effect
on resilience compared with low variance [var(a) = 0.05,
Fig. 2].
Like community size and interaction strength, the effects
of (a)symmetric interaction strengths and nestedness on
resilience were qualitatively robust, but their quantitative
effects on resilience did depend on and vary with another
structural property, namely degree (number of species
interactions). Asymmetric interaction strengths showed no
prominent effect on resilience except when the degree
exponent c was small (more negative values), in which case
2007 Blackwell Publishing Ltd/CNRS
212 T. Okuyama and J. N. Holland
Letter
Figure 3 Community resilience and structural properties of asymmetry of interactions strengths and community nestedness. (a) Asymmetry
of interaction strengths between a and b is measured by |a ) b| ⁄ max(a, b), where a large value indicates asymmetry, and 0 indicates perfect
symmetry. (b) Nestedness is reported as (100 ) T ) ⁄ 100, where T is matrix temperature; 1.0 is a completely nested community. Potential
interactions with degree distribution (c, the exponent for a modified degree distribution) are shown for both symmetry of interaction strength
(a) and nestedness (b).
there was a negative relationship with resilience (Fig. 3a). In
other words, for communities with small degree exponents,
that is lower connectivity, increasing asymmetry of interaction strengths reduced resilience. Similarly, resilience was
largely independent of nestedness except when degree (c)
values were small; in this case, there was a positive
relationship between resilience and highly nested communities of low connectivity (Fig. 3b). Indeed, nestedness of
empirical communities is high (0.86 on average, Table S2)
and interaction strengths appear to be highly asymmetric
(minimum estimate = 0.55, Table S2; but see Discussion),
implying that the stability of mutualistic communities in
nature may be sensitive to nestedness and asymmetric
interaction strengths. However, stability may not be strongly
influenced by either nestedness or asymmetric interaction
strengths, as empirical estimates of degree (c) indicate
intermediate to less negative values (mean c ± SE:
)1.46 ± 0.15; Table S2), rather than small (more negative)
values (c = )2.5) required of our analyses for such negative
and positive relationships to arise.
Community resilience showed a positive relationship with
degree, but its quantitative effects diminished with increases
in asymmetry of interaction strengths (Fig. 4). Increasing the
degree exponent c (less negative values; Figs 3 and 4),
corresponding with an increase in the mean number of
species interactions and a decrease in the disparity (variance)
among species in their degree, enhanced resilience, implying
that the stability of mutualistic communities increases with
community connectivity. This positive relationship between
resilience and degree was further enhanced by symmetric
(small asymmetry values) rather than asymmetric interaction
strengths, whether variance in interaction strength was low
or high (Fig. 4). Yet, when interaction strengths were
2007 Blackwell Publishing Ltd/CNRS
relatively symmetric (asymmetry = 0.19) and their variance
high, a parabolic relationship occurred between resilience
and degree (Fig. 4a). Because values of c affect both the
mean and variance of a communityÕs degree distribution, we
were unable to distinguish their relative effect on resilience.
Nonetheless, interplay between degree and other structural
properties (symmetry of interaction strengths, nestedness;
Figs 3 and 4) suggests that degree and community connectivity do play key roles in both the structure and dynamics of
mutualistic communities.
DISCUSSION
Central to the complexity–stability debate is reconciling the
standard paradox of negative relationships between community stability and species diversity, connectivity and
interaction strengths (May 1974). With the recent identification of some of their general topological and structural
properties (Bascompte et al. 2003, 2006a; Jordano et al. 2003;
Vázquez 2005; Guimarães et al. 2006, 2007; Ollerton et al.
2007; Vázquez et al. 2007), we have been able to conduct
some of the first analyses of how particular structures of
mutualistic communities affect their stability. Our results
indicate that mutualistic communities, like predator–prey
and competitive communities, may too entail intricate
relationships between structure and stability. Yet, contrary
to the negative complexity–stability relationships of food
webs that sparked the debate (May 1974), our initial analyses
show largely positive complexity–stability relationships for
mutualistic communities. In particular, stability was enhanced by increasing species diversity (Fig. 1); by increasing
community connectivity through a greater number of species
interactions (larger degree exponents; Figs 3 and 4); and by
Letter
Figure 4 Community resilience and structural properties of degree
distribution and interaction strengths. Degree is measured by c, the
exponent for a domain constrained degree distribution. Results are
reported for (a) high variance [i.e. var(a) = 1.0] and (b) low
variance [i.e. var(a) = 0.05] in interaction strengths. For each plot,
high, intermediate and low asymmetry values are based on the
simulation averages with q = )0.99, 0 and 0.99, respectively.
increasing the strength and symmetry of pairwise mutualistic
interactions (Figs 2–4). Collectively, our theoretical analyses,
guided by empirically described mutualistic communities,
indicate that stability is enhanced by large nested communities with many species interactions and strong, symmetric
interaction strengths. Mutualistic networks show great
variation in their structures, suggesting the importance of
such studies that delineate relationships between their
structural properties and community dynamics.
A key feature of mutualistic interactions that distinguishes
them from other interspecific interactions is their positive
feedback on population growth rates. Our treatment of this
positive feedback through nonlinear functional responses
differs fundamentally from historic (Gause & Witt 1935) and
more recent analyses (Bascompte et al. 2006a) with linear
functional responses. Our mutualistic communities with
Stability of mutualistic networks 213
saturating functional responses always had a stable equilibrium, while mutualistic communities with linear functional
responses have a locally stable equilibrium, only if interaction
strengths are weak and ⁄ or asymmetric (Bascompte et al.
2006a; see Discussion in Holland et al. 2006). As different
forms of functional responses may influence complexity–
stability relationships (Nunney 1980; but see Abrams &
Allison 1982), it is critical to use biologically appropriate
forms, otherwise misleading results may ensue. In spite of
recent claims to the contrary (Bascompte et al. 2006b), linear
functional responses are unlikely to occur for mutualistic
interactions (Holland et al. 2002; Thompson et al. 2006) and
consumer–resource interactions more generally, as they
apply only to a specific group (filter feeders) of organisms
(Jeschke et al. 2004). Moreover, animals foraging on
resources supplied by their mutualistic partners (e.g. plant–
pollinator interactions) are likely to entail handling times
(Ivey et al. 2003; Goyret & Raguso 2006), which is well
recognized to lead to saturating functional responses.
Because of such differences in equilibrium stability arising
from different forms of functional responses, we measured
stability through resilience, that is transient dynamics (return
rates) of mutualistic communities to their stable equilibrium,
whereas analyses of mutualistic communities with linear
functional responses have measured the qualitative stability
(stable or unstable) of equilibrium. Our results indicate that
strong, symmetric interaction strengths enhance resilience
(Figs 2–4), while weak and ⁄ or asymmetric interaction
strengths have been suggested to facilitate local equilibrium
stability of mutualistic communities with linear functional
responses (Bascompte et al. 2006a). Although these results
appear contradictory, different definitions of stability can
change conclusions drawn about the effects of an ecological
factor on stability (Chen & Cohen 2001). For the case at
hand, local equilibrium stability is lost for mutualistic
communities with linear functional responses when interaction strengths are not weak and ⁄ or asymmetric. Yet,
communities with linear functional responses exhibit the
same positive effect of strong, symmetric interaction
strengths on resilience, when they are examined within the
parameter range of the locally stable equilibrium.
Our theoretical result that strong, symmetric interaction
strengths enhance stability appears to be at odds with the
purported empirical finding that mutualistic communities
are often characterized by weak, asymmetric interaction
strengths (Bascompte et al. 2006a). Although precise definitions of interaction strength vary among empirical and
theoretical studies (Berlow et al. 2004; Wootton & Emmerson 2005), per capita measures of them have been
advocated, as they incorporate variation in species densities
and ultimately per capita measures underlie most other
concepts of interaction strength, including parameters of
theoretical models that they can be used to evaluate (Paine
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214 T. Okuyama and J. N. Holland
1992; Laska & Wootton 1998). Without some measure of
species densities (Wootton 1997; Holland et al. 2006),
empirical measures of the relative frequency of species
interactions (Bascompte et al. 2006a) do not correspond
with and are inadequate to estimate and evaluate predictions
of per capita interaction strengths of theoretical models of
mutualistic communities (Bascompte et al. 2006a; Holland
et al. 2006; this study). Although recent asymmetry indices
do effectively compare the congruence between interaction
strengths (Bascompte et al. 2006a; Vázquez et al. 2007), the
relative frequency metric within them assumes that the sum
of per capita interaction strengths is the same among species
(i.e. the summation constraint of the denominator of a
relative measure). Consider one species that interacts 100
times with each of two species and another species that
interacts 100 times with each of four species. While
frequency of interaction (100) is the same for both species,
relative frequencies of interaction differ, 0.5 (100 ⁄ 200) and
0.25 (100 ⁄ 400). In this way, the relative frequency of
interaction for any given species likely decreases with
increases in community size and its degree. For these
reasons, relative frequency of interaction is not an effective
measure of the asymmetry between species interaction
strengths or frequency distributions of interaction strengths
of mutualistic communities (i.e. ÔdependenciesÕ; Bascompte
et al. 2006a). Similar relative measures have been found to
be inadequate to describe interaction strength in other
consumer–resource systems (e.g. Paine 1980). Interaction
strength is an important concept in understanding species
interactions, but empirical descriptions of them are currently
lacking for mutualistic systems, thereby representing an
important area of future study (Wootton & Emmerson
2005).
Before discussing the implications of our results for
complexity–stability relationships of ecological systems as a
whole, we first address two potential caveats for the study of
complexity–stability relationships of mutualistic communities. First, the effects of structural properties on stability
may not necessarily be independent of one another, as the
same underlying biological processes responsible for one
structure (e.g. nestedness) may influence or vary with
another structure (e.g. community size and degree). For
example, even though we used the same algorithm in our
analyses, the range of observed nestedness among communities depended on their degree exponents, c (Fig. 3b). If
particular structures of ecological communities are studied
in isolation of others, empirical or theoretical analyses may
produce misleading conclusions about their effects on
community dynamics or the underlying biological mechanisms responsible for those structures (Santamarı́a &
Rodrı́guez-Gironés 2006).
Second, because we analysed resilience of Jacobian
matrices at equilibrium, relationships between structure
2007 Blackwell Publishing Ltd/CNRS
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and community density at equilibrium may contribute to
increased resilience because of the increased negative effects
of diagonal elements (intraspecific interactions) of matrices.
With this reasoning, community structures may have the
same correlation sign (positive or negative) with both
resilience and community density. For example, increasing
species diversity, the number of species interactions
(degree), and the strength and symmetry of interactions
did lead to monotonic increases in community density and
resilience, as each enhanced population growth due to
mutualism and return rates to the community equilibrium
following a perturbation. Yet, not all structural properties
had a strong effect on community density, nor did they
affect community density and resilience in a similar manner.
For example, while community density always increased
with degree c, the generally positive relationship between
resilience and degree shifted to a parabolic relationship
when interaction strengths were more symmetric and their
variance high (Fig. 4a). Thus, the resilience of mutualistic
communities does result to some degree from the positive
density-dependent feedback of mutualism on community
density, but the effects of each structural property on
resilience were not mediated solely through community
density, nor were they a mere artefact of model structure.
Rather, it is the biology of nonlinear functional responses of
mutualistic communities, i.e. their saturating positive feedback on population growth, which contributes to their
positive complexity–stability relationships.
We have shown that positive relationships occur
between the stability and structural properties of mutualistic
communities, including community size, nestedness, degree
(number of species interactions) and the strength and
symmetry of interspecific interactions. Yet, our studies
considered mutualistic communities in isolation of other
interspecific interactions. Although this simplifying assumption is a reasonable starting point, the more realistic
inclusion of other species interactions such as predation
and competition may be crucial to understanding the
influences of their structural properties. Moreover, when
mutualistic communities are stable and do not grow
unbounded, as suggested by this study, their role in
predator–prey food webs may be through the transient
dynamics. To this end, complexity–stability relationships of
food webs may too hinge on the inclusion of mutualistic
interactions. Contrary to the often-held belief that the
positive feedback of mutualism destabilizes food webs,
resolving the ongoing complexity–stability debate may be
aided by the recognition of the complementary effects of
mutualistic and predator–prey interactions on the stability
of ecological communities as a whole. For instance,
the stable equilibrium of mutualistic communities involving saturating functional responses (May 1981; Holland
et al. 2002, 2006; Thompson et al. 2006) may balance
Letter
predator–prey food webs that tend to oscillate and often
have non-equilibrium solutions (McCann et al. 1998;
McCann 2000; Rooney et al. 2006). Similarly, the positive
feedback of strong, symmetric mutualistic interactions that
enhances stability may complement the Ômany weak, few
strongÕ interactions and structural asymmetries that dampen
negative feedbacks and oscillations of predator–prey food
webs (McCann et al. 1998; Berlow 1999; McCann 2000;
Neutel et al. 2002; Rooney et al. 2006). Additionally, nested
mutualistic communities (Bascompte et al. 2003; Guimarães
et al. 2006, 2007; Lewinsohn et al. 2006; Ollerton et al. 2007)
may complement compartmentalized structures of antagonistic food webs (Krause et al. 2003; Lewinsohn et al. 2006;
Rooney et al. 2006). Given the ubiquity of mutualisms in
real food webs, it may well be that these and other
synergistic features of mutualistic and predator–prey
interactions collectively contribute to the stability of
ecological communities as a whole.
ACKNOWLEDGEMENTS
We thank J. Chase, D. DeAngelis, B. Fagan, D. Queller,
N. Waser and anonymous referees for their comments and
suggestions which greatly improved our analyses and paper.
This work was supported in part by the Rice Computational
Research Cluster funded by NSF under Grant CNS0421109. TO was supported by a Huxley Fellowship by
the Department of Ecology and Evolutionary Biology at
Rice University.
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SUPPLEMENTARY MATERIAL
The following supplementary material is available for this
article:
Appendix S1 Jacobian matrix of the model of mutualistic
communities.
Table S1 Symbols, definitions, and the range and default
values of model parameters.
Table S2 Empirical estimates of structural properties of
mutualistic networks.
This material is available as part of the online article
from: http://www.blackwell-synergy.com/doi/full/10.1111/
j.1461-0248.2007.01137.x
Please note: Blackwell Publishing are not responsible for the
content or functionality of any supplementary materials
supplied by the authors. Any queries (other than missing
material) should be directed to the corresponding author for
the article.
Editor, Jonathan Chase
Manuscript received 22 August 2007
First decision made 21 September 2007
Manuscript accepted 21 October 2007